| № |
Condition |
free/or 0.5$ |
| m82729 | Show that the open interval a, b and the closed interval a, b are both convex sets of R with the natural order (example 1.20). The hybrid intervals a, b and a, b are also convex. Show that intervals are the only convex sets in R. |
buy |
| m82730 | Show that the operator T: B(X) → B(X) defined by
is increasing. |
buy |
| m82735 | Show that the power functions f (x) = xn, n = 1, 2, . . . are convex on ℜ+. |
buy |
| m82738 | Show that the relation in example 1.18 is an order relation. That is, show that it is reflexive and transitive, but not symmetric.
Figure 1.6
Integer multiples |
buy |
| m82744 | Show that the representation in equation (6) is unique, that is, if
x = α1x1 + a2x2 +..........+ anxn
and also if
x = β1x1 + β2x2 +............+ βnxn
then αi = βi for all i. |
buy |
| m82745 | Show that the sequence of polynomials
converges uniformly on any compact subset S ⊆ ℜ. |
buy |
| m82748 | Show that the set L(X, Y) of all linear functions X → Y is a linear space. |
buy |
| m82750 | Show that the set of optimal strategies for each player is convex. |
buy |
| m82753 | Show that the Shapley value ϕ defined by (1) is linear. |
buy |
| m82757 | Show that the value function (example 2.28) can be alternatively defined by |
buy |
| m82759 | Show that the volume of the vat is maximized by devoting one-third of the material to the floor and the remaining two-thirds to the walls. |
buy |
| m82776 | Show that T(x*) ⊂ L(x*).
The Kuhn-Tucker first-order conditions are necessary for a local optimum at x* provided that the linearizing cone L(x*) is equal to the cone of tangents T(x*), which is known as the Abadie constraint qualification condition. The Kuhn-Tucker conditions follow immediately from proposition 5.3 by a straightforward application of the Farkas lemma. |
buy |
| m82783 | Show that
where Δ = det(A) = ad - bc. |
buy |
| m82787 | Show that x2 is convex on ℜ. |
buy |
| m82790 | Show that ∥y∥∞ satisfies the requirements of a norma on ℜ . |
buy |
| m82794 | Show the converse; that is, if f (x) =
then f .(x). 0 for every x ∈ S where S . {x ∈ X : gj(x) = 0 = j . 1; 2 . . .m}. |
buy |
| m82813 | Solve
subject to g(x) = x21 + x22 = 1 |
buy |
| m82815 | Solve the general Cobb-Douglas utility maximization problem
subject to p1x1 p2x2 +..........+ pnxn = m
[Follow the technique in example 5.23.] |
buy |
| m82816 | Solve the preceding problem starting from the hypothesis that xc > 0, xb = xd = 0. [If faced with a choice between xb > 0 and xd > 0, choose the latter.] |
buy |
| m82817 | Solve the problem |
buy |